Robust Design of Public Storage Warehouses
Yeming (Yale) Gong
EMLYON Business School
Rene de Koster
Rotterdam school of management, Erasmus University
Abstract
We apply robust optimization and revenue management in public storage
warehouses. We optimize the expected revenue of public storage warehouses
against the worst cases with a max-min revenue objective, and the decision
variables are mainly the number of storage units for each storage type. With the
robust design, we can observe worst-case revenue improvement.
1
Introduction
Public storage warehousing is a mature industry in the U.S. and a fast-growing
business in Europe and Asia Pacific. As the SSA (Self Storage Association, the
official association of this industry in the U.S.) reported, the number of such
facilities has climbed from only 6,601 facilities at year-end 1984 to over 46,500 at
year-end 2010. In the year of 2010, the total revenue of this industry in the U.S. was
$22 billion (USD). This business has become one of the fastest-growing sectors of
the United States commercial real estate industry over the last 35 years (see [1]).
The industry in Europe had boomed after a breathtaking growing from 2006 to
2009. According to the 2008 industry annual report of the SSA UK (Self Storage
Association of the UK, the official association of this industry in the UK), the
number of public storage warehouse facilities in Europe had sharply increased from
2007 to 2008, particularly in those emerging markets. For example, the industry had
increased by 200% in Poland, 117% in Switzerland, 64% in Denmark (see [2]).
Public storage warehousing usually operates as self-storage mode and thereby
its cost is low and stable. For public storage warehouses with an objective to
maximize the expected revenue and with a stable cost level, revenue management
rather than cost control is critical. A typical public storage warehouse contains
storage spaces of different storage types, each type with a specific number of
storage units. A customer rents one or multiple storage units of an appropriate type
for several months. However, the existing storage types or the available number of
storage units for each type may not fit the needs of the market. The number of
available storage units of some types may be insufficient, while other types are
abundant. This results in either lost customers and revenue, or inefficient utilization
of capacity of one type, which also may bring potential loss in another type.
The main facility design question is to provide a facility design improving the
expected revenue that is with a better fit between storage design (types and
numbers) and market demand. Based on data of warehouses in America, Europe and
Asia, Gong et al.[3] propose a facility design approach for public storage
warehouses with three different cases: an overflow customer rejection model, and
two models with customer upgrade possibilities: one with reservation and another
without reservation. They analyze models for real warehouse cases mainly by
stochastic models and dynamic programming (see [3], [4]).
While these methods assume a probability distribution or a specific stochastic
process of demands, it is difficult to accurately estimate the probability distribution
of the total arriving storage requests or associated revenue in some warehouses. To
provide a method to solve the practical problem, we present robust optimization
models to handle problems with uncertain marketing data and parameters which
cannot be accurately estimated.
Robust optimization is a developing literature body recently, which plays an
important role in revenue management (see [5]). Robust optimization is a method
for modeling optimization problems under uncertainty, to find optimal decisions for
the worst-case realization of the uncertainties within a given set. Typically, the
original uncertain optimization problem is converted into an equivalent
deterministic form (called the robust counterpart) using strong duality arguments
and then solved using standard optimization algorithms. Soyster [6], Ben-Tal and
Nemirovski [7], and Bertsimas and Sim [8] describe the method for constructing
robust counterparts for uncertainty sets of various structures. Robust optimization
has been applied in various fields including supply chain management, finance, and
logistics. Goh and Sim [8] develop ROME, an algebraic modeling toolbox designed
to solve a class of robust optimization problems in the MATLAB environment, to
solve application problems. AIMMS provides robust optimization add-on, with the
ability of handling with multi-periods optimization problems without falling into the
trap of “the curse of dimensionality”.
Revenue management (see [10]) is a main concern of public storage
warehouses. Robust optimization plays an important role in revenue management
(see [4]). We optimize the expected revenue of public storage warehouses against
the worst cases with a max-min revenue objective, and the decision variables are
mainly the number of storage units for each storage type. This research is mainly in
the field of facility design (see [11]). Rosenblat and Lee[12] propose a robust
approach to facility design. This paper makes contribution to facility planning (see
[11] ) and proposes a facility design method to handle risks and uncertainties and
with an objective to increase revenue.
2
Models
The main research problem is to determine how many storage type i units should
be constructed to fit demand. There are m different storage type units with a
specific integer storage size area ci ,1 ≤ i ≤ m . We assume that the random arrival
processes of customers requesting a storage type i unit, 1 ≤ i ≤ m are independent
Poisson processes with arrival rates λi and that the occupation times of a storage
type i unit are independent and identically distributed random variables with
expected occupation time β i . Customers requesting a storage type i unit are called
type i customers.
When a type i customer arrives and storage type i units are not fully occupied,
the customer will be accepted at a price of ri per period for a storage type i unit.
When all storage type i units are occupied, customers who initially are interested in
a storage type i unit, may accept with probability pi to pay a price of ri +1 for a
storage type i + 1 unit. A customer willing to accept this is called an upscaled
customer. If a storage type i + 1 unit is available, an upscaled customer will be
served, otherwise the customer is lost.
Let xi ,1 ≤ i ≤ m be the number of storage type i units built for type i customers at
level i and yi ,1 ≤ i ≤ m − 1 the type i + 1 units reserved for type i customers upscaled
to level i + 1 . A type i customer who finds upon arrival all the xi storage type i units
occupied may choose to be upscaled and use one of the yi reserved units, if one is
available. If all yi units are occupied, the customer is lost. Customers of type m
finding upon arrival all xm units busy are directly lost. A customer of type i + 1 is not
allowed to occupy one of the yi units reserved for upgraded type i customers. Each
rejected type i customer is willing to upscale with probability pi .
To analyze this model we first look at the process followed by each type of
customers separately. The long-run average revenue obtained from type i customers
can be split in the long-run average revenue obtained from the xi and yi units. The
number of occupied storage type i units among the available xi can again be
modeled by a queueing loss system. Due to our assumption of exponentially
distributed occupation times this is an M / M / xi / xi loss queue. The long-run
m
average revenue obtained from the xi units is given by
∑ r ρ (1 − B( x , ρ )) . The
i =1
i
i
i
i
long-run average revenue obtained from the yi units is
m −1
∑ rη
i =1
i i +1
( xi )β i (1 − Pr ( xi , yi ) ) , where ηi +1 ( xi ) = pi λi B ( xi , ρ i ) and Pr ( xi , yi ) the
rejection probability that an upscaled type i customer will find all the reserved
yi units occupied. Let Λ be the set of demand data for all storage types of
D quarters in the examining years, we calculate the worst performance as
m −1
⎡m
⎤
min λd ∈Δ ,d =1,...,D ⎢ ∑ ri ρ i (1 − B( xi , ρ i )) + ∑ riηi +1 ( xi )β i (1 − Pr ( xi , yi ) )⎥
i =1
⎣ i =1
⎦
Given the price ri for each storage type i unit per unit of time, the motivation of
robust design is to minimize the loss in revenue due to variation in demand, and to
decide how many units of each type to build (and reserve) such that we can find the
best one among the worst performances, and we present a robust model as follows,
m −1
⎧
⎡m
⎤⎫
max ⎨ min λd ∈Δ ,d =1,..., D ⎢ ∑ ri ρ i (1 − B( xi , ρ i )) + ∑ riηi +1 ( xi )β i (1 − Pr ( xi , yi ) )⎥ ⎬ ... ( R )
i =1
⎣ i =1
⎦⎭
⎩
m −1
s.t.
∑ (c x + c
i =1
i i
y ) + cm xm ≤ C
i +1 i
xi , yi ∈Z +
3
Analysis
We apply the following random search robust optimization algorithm to calculate
the design and solve model (R).
Algorithm: robust optimization for PS warehouses
(I) For t = 1: T .
( )
(I.1) Generate a sample x t from U x, x with consideration of the capacity
m
constraint
∑c x
i i
i =1
≤C.
(I.2)(A) For d = 1: D , compute the objective value Rd ( x t ) from the d th demand
pattern.
(I.2)(B)Compute the worst performance of this generated design x t in D demand
patterns R t = min ( Rd ( x t ), d = 1: D ) .
(II)Compute R ∗ = max ( R t , t = 1: T )
(III) Return x corresponding to R ∗ as the robust design.
We take the S.P. Rotterdam warehouse as an example, to illustrate how to
provide a robust design, such that a warehouse can achieve the best revenue
performance among the worst revenues from demand data of D quarters.
Table 1: Parameters of S.P. Rotterdam warehouse
Items
Types(m2)
Price (Euro)
Old design
Class 1
3
109
34
Class 2
6
132
44
Class 3
9
177
58
Class 4
12
225
25
Class 5
15
254
18
Class 6
18
372
27
Class 7
22
436
3
Class 8
27
468
4
We use optimal results from Table 3 in [3] , which is by a sensitivity analysis of
S.P. Rotterdam warehouse design based on data of 8 quarters, to construct the
search range ⎡( x, y ) , ( x, y )⎤ for problem (R). For iterations t = 1: T , we generate
⎣
⎦
t samples ( x, y ) ,τ = 1,..., t from the search range while considering the capacity
constraint. For each of the samples ( x, y )τ we compute its revenues for D different
τ
{
}
demand patterns, and obtain the worst value Rτ = min Rd ( ( x, y )τ , λd ) , d = 1,..., D for this
design sample. The optimal revenue corresponding to the robust design is now
R (t )∗ = max {Rτ ,τ = 1,..., t} for iteration t . We can then return the ( x, y ) value
corresponding to R(t )∗ as the robust design for iteration t . The theoretical optimal
robust revenue is R ∗ = lim R (t )∗ .
t →∞
We present the searching process {R(t )∗ , t = 1: T } in Fig.-1. By setting the
convergence criterion as R(t + 1) − R(t ) R(t ) ≤ 0.3% , we observe the search
converges in the last 500 iterations. We therefore calculate the average value from
the last 500 values {R (t )∗ , t = 2500 : 3000} , and obtain the robust optimal revenue
value as 43912, which is a significant improvement compared with the current
worst revenue value 32622 based on the current design. We use the design obtained
after 3000 iterations {50(6), 9(15), 19(16), 46(0), 2(15), 25(5), 9(4), 19(0)} as the
robust design.
Figure 1: Robust optimization for S.P. Rotterdam warehouse
4
Concluding remarks
This paper presents a robust design method for public storage warehouses by
applying robust optimization and revenue management. In regard to available
information, it needs to further consider two situations: (1) Some warehouses
maintain incomplete market data. They cannot accurately estimate customer
demand, including average order arrival rates. But they can roughly provide a scope
of customer demand by history data. This happens in small and middle facility
providers and some large-scale companies with relatively unqualified information
management. (2) Some warehouses can provide more market data. They can
estimate customer demand information, including average order arrival rates. They
cannot accurately specify some parameters of their market demand models; instead
they can roughly provide a scope of parameters by historical data. This happens in
some world-class facility providers (like Shurgard) and some middle-scale
companies with good information management systems.
Methods and technologies of robust optimization need improvement in the
further research. The limitation of random search robust optimization algorithm is
in searching speed and accuracy. It is interesting to explore better algorithms for
robust design of large–scale public storage warehouses.
Acknowledgements
Yeming Gong acknowledges the support from NSFC (70901028).
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